Degree of convergence of a function in generalized Zygmund space

Hare Krishna Nigam; Mohammad Mursaleen; Manoj Kumar Sah

doi:10.3934/mfc.2022029

Article Contents

2023, Volume 6, Issue 3: 484-499. Doi: 10.3934/mfc.2022029

This issue Previous Article Integral modification of Beta-Apostol-Genocchi operators Next Article Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator

Degree of convergence of a function in generalized Zygmund space

1.
Department of Mathematics, Central University of South Bihar, Gaya, Bihar, India
2.
Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan
3.
Department of Mathematics, Aligarh Muslim University, Aligarh, India

^*Corresponding author: M. Mursaleen
^*Corresponding author: M. Mursaleen

Received: February 2022

Revised: July 2022

Early access: August 2022

Published: August 2023

Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by

Abstract

In this paper, we obtain the results on the degree of convergence of a function of Fourier series in generalized Zygmund space using deferred Cesàro-generalized Nörlund $ (D^{h}_{g}N^{a,b}) $ transformation. Important corollaries are deduced from our main results. Some applications are also given in support of our main results.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Degree of convergence of function $ f $

Download: Full-size image PowerPoint slide

Figure 2. Degree of convergence of function $ f $

Download: Full-size image PowerPoint slide

Table 1. Degree of convergence of $f$ for different $n$

$n$	Degree of convergence of $f$
100	1.2575829
1000	1.1507909
10000	1.1082121
50000	1.0911295
100000	1.0854078
500000	1.0746045
1000000	1.0707630
.	.
.	.
$\infty$	1

| Show Table

DownLoad: CSV

Table 2. Degree of convergence of $f$ for different $n$

$n$	Degree of convergence of $f$
100	3.8368
1000	3.5991
10000	3.4794
50000	3.4274
100000	3.4096
500000	3.3759
1000000	3.3639
10000000	3.3315
100000000	3.3073
.	.
.	.
$\infty$	3.1416

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	R. P. Agnew, On deferred Cesàro means, Ann. Math., 33 (1932), 413-421. doi: 10.2307/1968524.
[2]	C. K. Chui, An Introduction to Wavelets, Wavelet Analysis and its Applications, 1. Academic Press, Inc., Boston, MA, 1992.
[3]	G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.
[4]	S. Lal, Approximation of functions belonging to the generalized Lipschitz class by $C_{1}N_{p}$ summability method of Fourier series, Appl. Math. Comput., 209 (2009), 346-350. doi: 10.1016/j.amc.2008.12.051.
[5]	S. Lal and A. Mishra, The method of summation $(E, 1)(N, p_{n})$ and trigonometric approximation of function in generalized Hölder metric, J. Indian Math. Soc., 80 (2013), 87-98.
[6]	B. A. London, Degree of Approximation of Hölder Continuous Functions, Thesis (Ph.D.)-University of Central Florida, 2008.
[7]	Det Kgl. Mollerup, Danske videnskabernes selska, Math.-Fys. Medd., 3 (1920).
[8]	H. K. Nigam, Degree of approximation of a function belonging to weighted $(L_{r}, \xi(t))$ class by $(C, 1)(E, q)$ means, Tamkang J. Math., 42 (2011), 31-37. doi: 10.5556/j.tkjm.42.2011.514.
[9]	H. K. Nigam and Md. Hadish, Approximation of a function in Hölder class using double Karamata $(K^ {\lambda, \mu})$ method, Eur. J. Pure Appl. Math., 13 (2020), 567-578. doi: 10.29020/nybg.ejpam.v13i3.3663.
[10]	H. K. Nigam and Md. Hadish, Best approximation of functions in generalized Hölder class, J. Inequal. Appl., (2018), Paper No. 276, 15 pp. doi: 10.1186/s13660-018-1864-y.
[11]	H. K. Nigam and Md. Hadish, Trigonometric approximation of functions by Hausdorff-Matrix product operators, Nonlinear Functional Analysis and Applications, 24 (2019), 675-689.
[12]	H. K. Nigam and S. Rani, Approximation of function in generalized Hölder class, Eur. J. Pure Appl. Math., 13 (2020), 351-368. doi: 10.29020/nybg.ejpam.v13i2.3667.
[13]	E. C. Titchmarsh, The Theory of Functions, Second edition, Oxford University Press, Oxford, 1939.
[14]	O. Töeplitz, Uberallagemeine lineara, Mittelbil. Dunger. P.M.F., 22 (2013), 113-119.
[15]	A. Zygmund, Trigonometric Series, 3rd rev. ed., Cambridge University Press, Cambridge, 2002.